Physics I- impossible question on oscillation (help needed).

[frasifrasi]

The question is

"The magnitude of the gravitational acceleration inside Earth is given approximately by g(r) = g_0(r/R_E), where g_0 is the surface value, r is the distance from Earth's center, and R_E is Earth's radius; the acceleration is directed toward Earth's center. Suppose a narrow hole were drilled straight through the center of Earth and out the other side. Neglecting air resistance, show that an object dropped into this hole executes simple harmonic motion, and find an expression for the period. Evalueate and compare with the period of a satellite in a circular orbit not far above Earth's surface."


I am gettinng that T = 2(pi)*sqrt(R_E/g_0)

How do I proceed from here? I am lost! I tried comparing and equations and think that R = RE, and their periods are exactly the same. is this correct?

 

[sokratesla]

I think your answer is correct. Have confidence in your reasoning. :-) Their periods are exact means that you found an interesting result!

First question is related to simple harmonic oscillation: [itex]\frac{d^2r}{dt^2}=-\frac{g_0}{R_E}r[/itex] where [itex]\omega=\sqrt{\frac{g_0}{R_E}}[/itex]. And second problem is related to constant circular motion: [itex]a=\frac{v^2}{R}[/itex], where [itex]a=g_0, v=\omega R_E, R=R_E[/itex] so same [itex]\omega[/itex]. But you should pat attention to the relation between angular velocity and frequency [itex]\omega=2\pi f[/itex]

 

[frasifrasi]

Ok, thanks. My work is very similar to yours--is that enough to answer the question? The reason I ask is because this is an end of chapter question, which is supposed to be one of the harder ones.

 

[sokratesla]

# It would enough for me if I were your teacher.

 

[frasifrasi]

How did you get that velocity?

isn't velocity supposed to be:

v^2 = v_0^2 + 2ax ?
= v = sqrt(Rg_0) ?

thanks and sorry about no latex.